\subsection{方法一}
\begin{tikzpicture}
    \coordinate[label=left:\textcolor{blue}{$A$}](A) at (0,0);
    \coordinate[label=right:\textcolor{blue}{B}](B) at (1.25, 1.25);

    \draw[draw=blue](A) -- (B);

    \draw[name path=D] let 
         \p1 = ($(B) - (A)$),
         \n2 = {veclen(\x1, \y1)}
       in
       (A) circle (\n2) 
       ;

    
    \draw[name path=E] let 
       \p1 = ($(B) - (A)$),
       \n2 = {veclen(\x1, \y1)}
     in 
     [name path=E](B) circle (\n2);

     \path[name intersections={of=D and E}];
     \coordinate[label=left:C](C) at (intersection-1);

     \draw(A) -- (C);
     \draw(C) -- (B);


\end{tikzpicture}
\subsection{方法二}

\begin{tikzpicture}
  \coordinate[label=left:\textcolor{blue}{$A$}](A) at (0,0);
  \coordinate[label=right:\textcolor{blue}{B}](B) at (1.25, 1.25);
  
  \draw (A) -- (B);

  \node(D)[circle through=(B), name path=D, draw=black, label=left:D] 
     at (A) {};
  \node(E)[circle through=(A), name path=E, draw=black, label=right:E] at (B) {};
  
  \path[name intersections={of=D and E}];
  \coordinate[label=left:C](C) at (intersection-1);

  \draw(A) -- (C);
  \draw(B) -- (C);
\end{tikzpicture}

\subsection{证明}
\begin{tikzpicture}[help lines/.style={draw=black!50, thin}, thick]
  \def \A{\textcolor{orange}{$A$}}
  \def \B {\textcolor{blue}{$B$}}
  \def \C {\textcolor{blue}{$C$}}
  \def \D {$D$}
  \def \E {$E$}
  \def \F {$F$}
  \def \G {$G$}
  \def \H {$H$}
  \def \K {$K$}
  \def \L {\textcolor{orange}{$L$}}
  
  \coordinate[label=\A](A) at (0, 0);
  \coordinate[label=\B](B) at ($(1, 0.2) + 0.1*(rand, rand)$);
  \coordinate[label=\C](C) at ($(1, 2) + 0.1*(rand, rand)$);

  \draw[help lines] (A) -- (B);
  \draw[blue](B) -- (C);
  
  \coordinate[label=above:\D] (D) at ($ (A)!0.5!(B) ! {sin(60)*2} ! 90:(B) $);
  \draw[help lines] (D) -- (A);
  \draw[help lines] (D) -- (B);
   
  \coordinate[label=\E](E) at ($(D)!3.75!(A)$);
  \coordinate[label=\F](F) at ($(D)!3.75!(B)$);

  \draw[help lines] (D) -- (E);
  \draw[help lines] (D) -- (F);

  \node[label=135:\H, circle through=(C), help lines, name path=H](H) at (B){};
  \path[name path=D--F](D) -- (F);
  \path[name intersections={of=H and D--F, by={[label=below:\G]G}}];

  \node[circle through=(G), help lines, name path=K, label=\K] at (D) {};

  \path[name path=D--E](D) -- (E);

  \path[name intersections={of=K and D--E, by={[label=below:\L]L}}];
  \draw[orange](A) -- (L);

  \foreach \x in {A, B, C, D, G, L}
    \filldraw [opacity=0.5, black](\x) circle (2pt); 



  

  


  

\end{tikzpicture}
